Zeta functions over zeros of zeta functions and an exponential-asymptotic view of the Riemann hypothesis. (English) Zbl 1365.11107
Summary: We review generalized zeta functions built over the Riemann zeros (in short: ‘superzeta’ functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the Keiper-Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann hypothesis being true or not, and the outcome is a sharp exponential-asymptotic criterion for the Riemann hypothesis that only refers to the large-order Keiper-Li coefficients. As a new result, that criterion, then Li’s criterion, are transposed to a novel sequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1 for Keiper-Li).
MSC:
11M41 | Other Dirichlet series and zeta functions |
11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
11M35 | Hurwitz and Lerch zeta functions |
30B40 | Analytic continuation of functions of one complex variable |
30E15 | Asymptotic representations in the complex plane |
41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |